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  • European Journal of Applied Mathematics, Volume 1, Issue 4
  • December 1990, pp. 371-387

Uniqueness and stability of the solution to a thermoelastic contact problem

  • Peter Shi (a1) and Meir Shillor (a1)
  • DOI:
  • Published online: 01 July 2009

Uniqueness and continuous dependence on the initial temperature are proved for a onedimensional, quasistatic and frictionless contact problem in linear thermoelasticity. First the problem is reformulated in such a way that it decouples. The resulting problem for the temperature is a nonlinear integro-differential equation. Once the temperature is known the displacement is recovered from an appropriate variational inequality. Uniqueness is proved by considering an integral transform of the temperature. The steady solution is obtained and the asymptotic stability is shown. It turns out that the asymptotic behaviour and the steady state are determined by a relation between the coupling constant a and the initial gap.

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J. R. Barber 1978 Contact problems involving a cooled punch. J. Elasticity 8, 409423.

J. R. Barber & T. Zhang 1988 Transient behavior and stability for the thermoelastic contact of two rods of dissimilar materials. Int. J. Mech. Sci. 30, 691706.

M. Comninou & T. Dundurs 1980 On lack of uniqueness in heat conduction through a solid to solid contact. J. Heat Transfer 102, 319323.

W. A. Day 1981 Justification of the uncoupled and quasistatic approximations in a problem of dynamic thermoelasticity. Arch. Rat. mech. Anal. 77, 387396.

W. A. Day 1982 Further justification of the uncoupled and quasistatic approximations in thermoelasticity. Arch. Rat. Mech. Anal. 79, 8595.

W. A. Day 1985 Heat Conduction within Linear Thermoelasticity. Springer, New York.

G. Duvaut & J. L. Lions 1972 Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Rat. Mech. Anal. 46, 241279.

G. Duvaut & J. L. Lions 1976 Inequalities in Mechanics and Physics. Springer, Berlin.

O. A. Ladyzhenskaya 1985 The Boundary Value Problems of Mathematical Physics. Springer, New York.

J. L. Lions & E. Magenes 1972 Non-Homogeneous Boundary Value Problems and Applications. Vol. 11. Springer, New York.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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